J. Leaños1, C. Ndjatchi-Mbe-Koua2, L. M. Rivera-Martínez3
1Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, México jleanos@mate.reduaz.mx
2Coordinación de Ingeniería Mecatrónica y Energía, Universidad Politécnica de Zacatecas, México ndjatchi@upz.edu.mx
3Faculty of Mathematics, University of Vienna, Austria luismanuel.rivera@univie.ac.at
Abstract. In [4] it was proved that, if a simple Euclidean arrangement of pseudolines has no(≥5)-gons, then it is stretchable. In the opposite direction, it is known that not every simple Euclidean arrangement with three(≥5)-gons is stretchable; see [8]. Thus the following question arises naturally: Are the simple Euclidean arrangements with one or two(≥5)-gons stretchable? In this paper we give a large class of stretchable simple Euclidean arrangements with one(≥5)-gon. More precisely, we prove that, ifLis a Euclidean arrangement ofnpseudolines with one n-gon, thenLis stretchable. We also prove that the number of such arrangements isΩ(2n/2).
Introduction
A simple noncontractible closed curve in the projective plane P is a pseudoline, and an arrangement of pseudolines is a collection B = {p0, p1, . . . , pn} of pseudolines that intersect (necessarily cross) pairwise exactly once. Since P\p0 is homeomorphic to the Euclidean plane E, we may regard {p1, . . . , pn} as an arrangement of pseudolines in E (and regardp1, . . . , pnaspseudolinesinE). An arrangement issimpleif no point belongs to more than two pseudolines. The cell complex of an Euclidean arrangement in P has both bounded and unbounded cells. As in [4], we are only interested in bounded cells (whose interiors are thepolygons or faces). Thus it is clear what is meant by a triangle, a quadrilateral, or, in general, an n-gon of the arrangement. Any m-gon with m ≥ nis called a(≥n)-gon.
Anarrangement of linesinEis a collection of straight lines, no two of them parallel.
Thus, every arrangement of lines is an arrangement of pseudolines. On the other hand, not every arrangement of pseudolines isstretchable, that is, equivalent to an arrangement of lines, where two arrangements are equivalentif they generate isomorphic cell decom-positions of E. Every arrangement of eight pseudolines is stretchable [3], but there is a simple non-stretchable arrangement (Figure 1) of nine pseudolines [8], which is unique up to isomorphism [3]. Since the arrangement in Figure 1 is non-stretchable and has only three(≥5)-gons, it follows that not every simple Euclidean arrangement with three (≥5)-gons is stretchable. In the opposite direction, in [4] was proved that every simple Euclidean arrangement with no(≥5)-gons is strechable. With these facts in mind, it is natural to ask what is the role of the(≥5)-gons in the stretchability of simple Euclidean
3Partially supported by an ERC Grant.
CRM Documents, vol.8, Centre de Recerca Matemàtica, Bellaterra (Barcelona), 2011
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130 Euclidean arrangements ofnpseudolines with onen-gon are stretchable
arrangements. This work is a first effort in this direction. In particular, here we study the stretchability of certain class of Euclidean arrangements with one(≥5)-gon.
Stretchability questions are typically difficult: deciding stretchability is NP-hard [9]
even for simple arrangements [10]. The concept of stretchability is particularly relevant because of the close connection between arrangements of pseudolines and rank 3 oriented matroids: on this ground, the problem of stretchability of arrangements is equivalent to the problem of realizability for oriented matroids (see [1, 7]).
Let=denote the set of Euclidean arrangements ofn pseudolines with onen-gon. It follows from the definition of=that every arrangement of=must be simple. Our aim is to prove that every element of= is stretchable and that the number of non-isomorphic arrangements of=withm pseudolines grows exponentially withm.
Figure 1. A simple non-stretchable arrangement with three(≥5)-gons.
1 Results
Our main results are the following:
Theorem 1.1. If L is an arrangement of=, then L is stretchable.
Theorem 1.2. The number of non-isomorphic Euclidean arrangements ofnpseudolines with one n-gon is Ω(2n/2).
To prove Theorem 1.1, we need two lemmas. Lemma 1.4 is a consequence of the proof of Proposition 1.2 in [2] and the definition of =. Lemma 1.5 establishes structural properties of the elements of=. Lemmas 1.3 and 1.5 are both easy to see.
Since Theorem 1.1 is trivial forL ∈ =with|L| ≤4, we may assume that|L| ≥5.
Lemma 1.3. If L ∈ =, then the|L|-gon ofL is the unique(≥5)-gon of L.
Lemma 1.4. If L ∈ =, then the |L|-gon of L has at most two edges that are adjacent to an unbounded cell of L.
Lemma 1.5. LetL ∈ =. If an edgeeof the |L|-gon ofLis not adjacent to an unbounded cell of L, then eis also an edge of a triangle of L.
We now give a sketch of the proof of Theorem 1.1. See [5] for further details.
1.1 Proof of Theorem 1.1
The proof is by induction on |L|. In [3] it was shown that any Euclidean arrangement with 8 pseudolines is stretchable. So we may assume that |L| = n ≥9 and that every arrangement of = with j < n pseudolines is stretchable. LetP be then-gon of L. By Lemma 1.4,P has at most two edges that are adjacent to an unbounded cell ofL. Hence,
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P has three consecutive edges, saya0,b0 andc0, which are not adjacent to an unbounded cell ofL. By Lemma 1.5, each ofa0,b0 andc0 is an edge of a triangle of L. Let A,B and C be, respectively, the triangles ofLthat are adjacent to a0,b0 andc0. Let a,b andc be the supporting pseudolines ofa0,b0andc0, respectively. Assume without loss of generality that a, b and c are directed in such a way that P lies to the left of each of them. See Figure 2.
Forx∈ {a, b, c}and `= 1, . . . , n−1, let x` be the`-th pseudoline ofL \ {x} that is crossed byx. SinceL is simple, the labelsa1, . . . , an−1;b1, . . . , bn−1; andc1, . . . , cn−1 are well-defined. We assume from now on thatak=c,bt=aandcr=a.
LetX,Y,W and Z be the four unbounded regions defined byaandc(see the small drawing in Figure 2). Note that P ⊂ X. Since P has an edge in every pseudoline of L, thenZ (respectively,Y) contains no crossings between pseudolines of {a1, . . . , ak−2}, (respectively,{cr+2, . . . , cn−1}). Using these facts, we can obtain Equations (1) and (2) by a straightforward argument.
bt−j =
(ak−1−j =cr−j if j= 1,2, . . . , r−1, ak−1−j if j=r, r+ 1, . . . , t−1.
(1)
bt+1+i =
(cr+1+i =ak+i if i= 1,2, . . . , n−k−1,
cr+1+i if i=n−k, n−k+ 1, . . . , n−t−2.
(2)
From Equations (1) and (2) it follows thatL looks like in Figure 2.
a
b c
a’ b’ c’
X W
Z Y
c a
A B C
P
Figure 2. The structure of an arrangement of=.
Clearly, L \ {b} belongs to = and, by induction, it is stretchable. Let L∗b be an arrangement of straight lines, which is equivalent to L \ {b}. If θ denotes an element of L \ {b}, we denote by θ∗ the corresponding element in L∗b. For s= 1, . . . , k−r−1, let ms be the slope of a∗s. Let H = {a1, . . . , ak−r−1} (the thin pseudolines in Figure 2 are the elements of H). Since any two lines of L∗b intersect exactly once, L∗b has no lines with equal slopes. Moreover, since the crossing of any two lines of H∗ is in X∗, m0 < m1 <· · ·< mk−r−1 < mk−r, wherem0 andmk−r are, respectively, the slopes ofa∗ andc∗. Letd∗ be the line with slope(mk−t−1+mk−t)/2 through v=a∗∩c∗. Since L∗b andL\{b}are equivalent andm0 < m1 <· · ·< mk−r,d∗ crosses the lines ofL∗b\{a∗, c∗} in the exact same order in whichbcrosses the pseudolines ofL \ {a, b, c}. Also, note that d∗ can be perturbed in such a way that: (i) the order in which d∗ crosses the lines of L∗b \ {a∗, c∗} is preserved, and (ii) d∗ intersectsX∗. Finally, note that the arrangement of lines obtained by such a perturbation ofd∗ is equivalent to L, as desired.
132 Euclidean arrangements ofnpseudolines with onen-gon are stretchable
1.2 Proof of Theorem 1.2
A2-colored necklace with2mbeads in which opposite beads have different colors is a self-dual necklace. An example of a self-dual necklace is shown in Figure 3i). Two self-dual necklaces are isomorphic if one can be obtained from the other by rotation or reflection or both. In [6] it was proved that the number of non-isomorphic self-dual necklaces is Ω(2m/2). So, it is enough to exhibit a one-to-one correspondence between the set of self-dual necklaces and a subset of =. LetC be a self-dual necklace with2m≥6 beads colored 0 and 1, and let P be the regular polygon of 2m sides. Now we extend every edge ofP to both sides in such a way that each pair of non-parallel segments intersect.
See Figure 3 ii). Finally, we intersect every pair of parallel segments according to color 1ofC as shown in Figure 3 iii). By construction, the resultant arrangementPC belongs to =. It is easy to see that distinct necklaces generate distinct arrangements.
P
C 0
1 1
0 1
0 0 1
iii) ii)
i)
C P
Figure 3. How to associate an arrangement of=to each self-dual necklace with 2mbeads.
Remark 1.6. An anonymous referee has observed that the elements of=are nothing but the extensions of the cyclic arrangement by one new line that does not cross the n-gon and consequently the number of non-isomorphic Euclidean arrangements ofnpseudolines with one n-gon is Ω(2n−1/n).
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